For sparse signal recovery, similar comments to that of Section REF  can be made to PGROTP.
The discussion is omitted here.
Before we close this section, we list a few RIC conditions in terms of \(\delta _{3k}\)  for the proposed algorithms with different \(q\) , i.e., \(q=2k,3k,4k\) .
The results shown in Table 1 are derived based on (REF ), (REF ) and (REF ) for the \(q\)  as given above, respectively.
It is worth mentioning that, when \(q=n\) , the partial gradient \({\cal H}_q(\nabla f(x))\)  becomes the full gradient \(\nabla f(x)\) , and the algorithms in this paper are reduced to the optimal \(k\) -thresholding (OT), relaxed optimal \(k\) -thresholding (ROT) and relaxed optimal \(k\) -thresholding pursuit (ROTP) algorithm, respectively.
The sufficient conditions for the convergence of these algorithms were studied in [1]}, [2]}.
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